A positional control problem for a nonlinear parabolic system (Q5942096)

This is a continuation of previous work by the same author. The problem consists in determining the so-called guaranteed control, where the phase trajectory ends in a previously designated set. In this case the system describes the solidification of a material. In addition to the control, the author also considers an uncontrolled perturbation. Specifically, the domain \(\Omega\subset \mathbb{R}^n\), \(n= 2\) or \(3\), with sufficiently smooth boundary describes the shape of the material. A nonlinear system of coupled equations describing the process of solidification of a cooled fluid shape was originally introduced by the author in [ibid. 28, 961--967 (1992; Zbl 0837.93032) and ibid. 27, 597--603 (1991; Zbl 0785.93050)]. The earlier work on this type of problem was done by \textit{Yu. S. Osipov} and collaborators [see for example Sov. Math., Dokl. 12, 262--266 (1971); translation from Dokl. Akad. Nauk SSSR 196, 779--782 (1971; Zbl 0229.90067), J. Appl. Math. Mech. 35, 92--99 (1971); translation from Prikl. Mat. Mekh. 35, 123--131 (1971; Zbl 0254.90069)]. Because of a cubic nonlinearity, the \(H'(\Omega)\) space is embedded in \(L_6\). After a brief statement describing the feedback control under incomplete information, the author sets out to prove its existence. It is stated in the classical formalism: Given an \(\varepsilon> 0\) there exists an \(h_0(\varepsilon)\) such that etc., describing the fact that the state of the system ends in the prescribed admissible set. The proof of this existence theorem requires first the proofs of several highly non-trivial inequalities, using the Lions-Magenes kind of arguments, plus some tricks of the trade. The algorithm described by the author is stable under a priori described perturbations. This is a difficult project requiring familiarity with the modelling of industrial solidification processes, yet at the same time following faithfully the tradition of J.-L. Lions in maintaining functional analytic rigor.